Particle spectra of general Ricci-type Palatini or metric-affine theories

In the context of weak-field metric-affine (i.e. Palatini) gravity near Minkowski spacetime, we compute the particle spectra in the simultaneous presence of all independent contractions quadratic in Ricci-type tensors. Apart from the full metric-affine geometry, we study kinematic limits with vanishing torsion (i.e. a symmetric connection) and vanishing non-metricity (i.e. a metric connection, which is physically indistinguishable from Poincar\'e gauge theory at the level of the particle spectrum). We present a detailed report on how spin-parity projection operators can be used to derive systematically and unambiguously the character of the propagating states. The unitarity constraints derived from the requirements of tachyon- and ghost-freedom are obtained. We show that, even in the presence of all Ricci-type operators, only a narrow selection of viable theories emerges by a tuning.